9,765 research outputs found
Higher-order infinite horizon variational problems in discrete quantum calculus
We obtain necessary optimality conditions for higher-order infinite horizon
problems of the calculus of variations via discrete quantum operators.Comment: Submitted 11-May-2011; revised 16-Sept-2011; accepted 02-Dec-2011;
for publication in Computers & Mathematics with Application
Discrete-Time Fractional Variational Problems
We introduce a discrete-time fractional calculus of variations on the time
scale , . First and second order necessary optimality
conditions are established. Examples illustrating the use of the new
Euler-Lagrange and Legendre type conditions are given. They show that solutions
to the considered fractional problems become the classical discrete-time
solutions when the fractional order of the discrete-derivatives are integer
values, and that they converge to the fractional continuous-time solutions when
tends to zero. Our Legendre type condition is useful to eliminate false
candidates identified via the Euler-Lagrange fractional equation.Comment: Submitted 24/Nov/2009; Revised 16/Mar/2010; Accepted 3/May/2010; for
publication in Signal Processing
Calculus of Variations on Time Scales and Discrete Fractional Calculus
We study problems of the calculus of variations and optimal control within
the framework of time scales. Specifically, we obtain Euler-Lagrange type
equations for both Lagrangians depending on higher order delta derivatives and
isoperimetric problems. We also develop some direct methods to solve certain
classes of variational problems via dynamic inequalities. In the last chapter
we introduce fractional difference operators and propose a new discrete-time
fractional calculus of variations. Corresponding Euler-Lagrange and Legendre
necessary optimality conditions are derived and some illustrative examples
provided.Comment: PhD thesis, University of Aveiro, 2010. Supervisor: Delfim F. M.
Torres; co-supervisor: Martin Bohner. Defended 26/July/201
Euler-Lagrange equations for composition functionals in calculus of variations on time scales
In this paper we consider the problem of the calculus of variations for a
functional which is the composition of a certain scalar function with the
delta integral of a vector valued field , i.e., of the form
. Euler-Lagrange
equations, natural boundary conditions for such problems as well as a necessary
optimality condition for isoperimetric problems, on a general time scale, are
given. A number of corollaries are obtained, and several examples illustrating
the new results are discussed in detail.Comment: Submitted 10-May-2009 to Discrete and Continuous Dynamical Systems
(DCDS-B); revised 10-March-2010; accepted 04-July-201
Necessary Optimality Conditions for Higher-Order Infinite Horizon Variational Problems on Time Scales
We obtain Euler-Lagrange and transversality optimality conditions for
higher-order infinite horizon variational problems on a time scale. The new
necessary optimality conditions improve the classical results both in the
continuous and discrete settings: our results seem new and interesting even in
the particular cases when the time scale is the set of real numbers or the set
of integers.Comment: This is a preprint of a paper whose final and definite form will
appear in Journal of Optimization Theory and Applications (JOTA). Paper
submitted 17-Nov-2011; revised 24-March-2012 and 10-April-2012; accepted for
publication 15-April-201
The contingent epiderivative and the calculus of variations on time scales
The calculus of variations on time scales is considered. We propose a new
approach to the subject that consists in applying a differentiation tool called
the contingent epiderivative. It is shown that the contingent epiderivative
applied to the calculus of variations on time scales is very useful: it allows
to unify the delta and nabla approaches previously considered in the
literature. Generalized versions of the Euler-Lagrange necessary optimality
conditions are obtained, both for the basic problem of the calculus of
variations and isoperimetric problems. As particular cases one gets the recent
delta and nabla results.Comment: Submitted 06/March/2010; revised 12/May/2010; accepted 03/July/2010;
for publication in "Optimization---A Journal of Mathematical Programming and
Operations Research
Direct and Inverse Variational Problems on Time Scales: A Survey
We deal with direct and inverse problems of the calculus of variations on
arbitrary time scales. Firstly, using the Euler-Lagrange equation and the
strengthened Legendre condition, we give a general form for a variational
functional to attain a local minimum at a given point of the vector space.
Furthermore, we provide a necessary condition for a dynamic
integro-differential equation to be an Euler-Lagrange equation (Helmholtz's
problem of the calculus of variations on time scales). New and interesting
results for the discrete and quantum settings are obtained as particular cases.
Finally, we consider very general problems of the calculus of variations given
by the composition of a certain scalar function with delta and nabla integrals
of a vector valued field.Comment: This is a preprint of a paper whose final and definite form will be
published in the Springer Volume 'Modeling, Dynamics, Optimization and
Bioeconomics II', Edited by A. A. Pinto and D. Zilberman (Eds.), Springer
Proceedings in Mathematics & Statistics. Submitted 03/Sept/2014; Accepted,
after a revision, 19/Jan/201
The Variational Calculus on Time Scales
The discrete, the quantum, and the continuous calculus of variations, have
been recently unified and extended by using the theory of time scales. Such
unification and extension is, however, not unique, and two approaches are
followed in the literature: one dealing with minimization of delta integrals;
the other dealing with minimization of nabla integrals. Here we review a more
general approach to the calculus of variations on time scales that allows to
obtain both delta and nabla results as particular cases.Comment: 15 pages; Published in: Int. J. Simul. Multidisci. Des. Optim. 4
(2010), 11--2
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